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Perfect Zero-Knowledge PCPs for #P

Tom Gur, Jack O'Connor, Nicholas Spooner

TL;DR

This work constructs perfect zero-knowledge probabilistically checkable proofs (PZK-PCPs) for every language in #P and introduces the notion of locally simulatable encodings: randomised encodings in which every local view of the encoding can be efficiently sampled given a local view of the message.

Abstract

We construct perfect zero-knowledge probabilistically checkable proofs (PZK-PCPs) for every language in #P. This is the first construction of a PZK-PCP for any language outside BPP. Furthermore, unlike previous constructions of (statistical) zero-knowledge PCPs, our construction simultaneously achieves non-adaptivity and zero knowledge against arbitrary (adaptive) polynomial-time malicious verifiers. Our construction consists of a novel masked sumcheck PCP, which uses the combinatorial nullstellensatz to obtain antisymmetric structure within the hypercube and randomness outside of it. To prove zero knowledge, we introduce the notion of locally simulatable encodings: randomised encodings in which every local view of the encoding can be efficiently sampled given a local view of the message. We show that the code arising from the sumcheck protocol (the Reed-Muller code augmented with subcube sums) admits a locally simulatable encoding. This reduces the algebraic problem of simulating our masked sumcheck to a combinatorial property of antisymmetric functions.

Perfect Zero-Knowledge PCPs for #P

TL;DR

This work constructs perfect zero-knowledge probabilistically checkable proofs (PZK-PCPs) for every language in #P and introduces the notion of locally simulatable encodings: randomised encodings in which every local view of the encoding can be efficiently sampled given a local view of the message.

Abstract

We construct perfect zero-knowledge probabilistically checkable proofs (PZK-PCPs) for every language in #P. This is the first construction of a PZK-PCP for any language outside BPP. Furthermore, unlike previous constructions of (statistical) zero-knowledge PCPs, our construction simultaneously achieves non-adaptivity and zero knowledge against arbitrary (adaptive) polynomial-time malicious verifiers. Our construction consists of a novel masked sumcheck PCP, which uses the combinatorial nullstellensatz to obtain antisymmetric structure within the hypercube and randomness outside of it. To prove zero knowledge, we introduce the notion of locally simulatable encodings: randomised encodings in which every local view of the encoding can be efficiently sampled given a local view of the message. We show that the code arising from the sumcheck protocol (the Reed-Muller code augmented with subcube sums) admits a locally simulatable encoding. This reduces the algebraic problem of simulating our masked sumcheck to a combinatorial property of antisymmetric functions.
Paper Structure (36 sections, 32 theorems, 67 equations, 1 figure, 1 algorithm)

This paper contains 36 sections, 32 theorems, 67 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our results
  3. Acknowledgements
  4. Techniques
  5. Zero knowledge IOPs.
  6. Structure versus randomness
  7. Key challenge: local simulation of combinatorial and algebraic structure.
  8. Combinatorial structure of antisymmetric functions
  9. Local simulation of random low-degree extensions
  10. Local simulation of subcube sums of LDEs
  11. Preliminaries
  12. Coding theory
  13. Linear-algebraic claims
  14. The combinatorial nullstellensatz
  15. Locally simulatable encodings
  16. ...and 21 more sections

Key Result

Theorem 1

$\#\mathsf{P} \subseteq \mathsf{PZK}\text{-}\mathsf{PCP}[\mathrm{poly},\mathrm{poly}]$.

Figures (1)

  • Figure 1: A general antisymmetric matrix ($n = 5$), with an element of the dual highlighted ($r = 2, t = 7$).

Theorems & Definitions (115)

  • Theorem 1: Informally stated, see \ref{['thm:main']}
  • Remark 1.1
  • Theorem 2: informal, see \ref{['thm:rm-constraint-detector']}
  • Lemma 1
  • Theorem 2.1: Informally stated, see \ref{['theorem:sigma-rm-dual-z']}
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5: Zero codes
  • Definition 3.6: Sum codes
  • ...and 105 more